New local characterizations of the weighted energy class $\mathcal{E}_{\chi,\mathrm{loc}}(\Omega)$

Let $\Omega\subset\mathbb{C}^n$ be a hyperconvex domain and let $\chi:\mathbb{R}^-\to\mathbb{R}^+$ be a decreasing function. This note studies the local weighted energy class $\mathcal{E}_{\chi,\mathrm{loc}}(\Omega)$ introduced in \cite{HHQ13}. We establish two main results on local membership in this class. First, we prove a new local boundedness property for the weighted Monge--Amp\`ere energy: if $u\in\mathrm{PSH}^-(\Omega)$ admits suitable local majorants in $\mathcal{E}_{\chi,\mathrm{loc}}$ near the boundary of every relatively compact hyperconvex subdomain $D\Subset\Omega$, then the weighted energy $\int_K \chi(u)(dd^c u)^n$ remains locally finite for every compact set $K\subset D$. This gives the first explicit local control of the energy functional and is new even in the unweighted setting. Second, we obtain a substantial improvement concerning the local control of the Monge--Amp\`ere measure. We show that if, in addition to the boundary condition, $(dd^c u)^n$ is locally dominated by $(dd^c w)^n$ for some $w\in\mathcal{E}_{\chi,\mathrm{loc}}(D)$ inside $D$, then $u\in\mathcal{E}_{\chi,\mathrm{loc}}(D)$. This domination condition is strictly weaker than the previous requirement of local finiteness of the weighted energy, thereby significantly enlarging the class of admissible functions. Our results extend and refine the local theory developed in \cite{Q24,Q25} and provide a more flexible framework for plurisubharmonic functions with possible singularities on compact subsets.